Spectral theory of random cusped hyperbolic surfaces

The aim of this thesis is to study the spectral theory of random finite-area noncompact hyperbolic surfaces, focusing on the spectral gap. We study the size of the spectral gap for two different models of random surfaces: random covers and the Weil-Petersson model. First we show that for any non-compact finite-area hyperbolic surface X, there is a constant C > 0 such that a uniformly random degree-n cover X_n has no eigenvalues below 1/4−C(logloglogn)^2/log log n , other than those of X, with probability tending to 1 as n → ∞. Secondly, we show that for any ε > 0, α ∈ [0, 1 2), as g → ∞ a generic finite-area genus g hyperbolic surface with n = O(gα) cusps, sampled with probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below 1/4 − ((2α+1)/4)^2 −ε. For α = 0 this gives a spectral gap of size 3/16 −ε and for any α < 1/2 gives a uniform spectral gap of explicit size.